### Abstract

The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type II functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.

Original language | English |
---|---|

Pages (from-to) | 4314-4338 |

Number of pages | 25 |

Journal | Mathematical Biosciences and Engineering |

Volume | 16 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

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### Keywords

- Asymptotic expansions
- Epidemic models
- Geometrical singular perturbation
- Quasi steady state assumption
- Seasonally-forced models
- Vector borne disease dynamics

### Cite this

*Mathematical Biosciences and Engineering*,

*16*(5), 4314-4338. https://doi.org/10.3934/mbe.2019215

}

*Mathematical Biosciences and Engineering*, vol. 16, no. 5, pp. 4314-4338. https://doi.org/10.3934/mbe.2019215

**On the role of vector modeling in a minimalistic epidemic model.** / Rashkov, Peter; Venturino, Ezio; Aguiar, Maira; Stollenwerk, Nico; Kooi, Bob W.

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - On the role of vector modeling in a minimalistic epidemic model

AU - Rashkov, Peter

AU - Venturino, Ezio

AU - Aguiar, Maira

AU - Stollenwerk, Nico

AU - Kooi, Bob W.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type II functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.

AB - The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type II functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.

KW - Asymptotic expansions

KW - Epidemic models

KW - Geometrical singular perturbation

KW - Quasi steady state assumption

KW - Seasonally-forced models

KW - Vector borne disease dynamics

UR - http://www.scopus.com/inward/record.url?scp=85066135670&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066135670&partnerID=8YFLogxK

U2 - 10.3934/mbe.2019215

DO - 10.3934/mbe.2019215

M3 - Article

VL - 16

SP - 4314

EP - 4338

JO - Mathematical Bioscience and Engineering

JF - Mathematical Bioscience and Engineering

SN - 1547-1063

IS - 5

ER -